Supplemental document accompanying submission to Optics Express
Title: In situ detection of electrochemical reaction by weak measurement Authors: Le Liu, Zhangyan Li, Yang Xu, Kaijie Ma, Jingyu Xi, Tian Guan, Fuying Li, ZhouChongqi, suyi zhong, Yonghong He Submitted: 3/28/2021 10:39:39 PM
Supplementary Information:
In situ detection of electrochemical reaction by weak measurement
ZHANGYAN LI,1-3, △ YANG XU,3, △ KAIJIE MA,4, LE LIU,1,* JINGYU XI,1 TIAN
GUAN,3,* FUYING LI,1 CHONGQI ZHOU,2,3, SUYI ZHONG,3 AND YONGHONG
HE,2,3
1 Institute of Green Chemistry and Energy, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China. 2 Department of Physics, Tsinghua University, Beijing 100084, China. 3 Shenzhen Key Laboratory for Minimal Invasive Medical Technologies, Institute of Optical imaging and Sensing, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China. 4 Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan △ Both the authors contributed equally to this work * [email protected] * [email protected]
1. The weak measurement theory
In the quantum weak measurement theory, the quantum state of system to be detected is firstly selected, which is called the pre-selected state. Next, the system to be detected is weakly coupled with the measuring device, and the strength of the coupling is not strong enough to cause the quantum state of system to collapse. The third step is to make another selection and obtain the post-selected state. When the pre-selected state and the post-selected state are nearly orthogonal, a measurement value called weak value can be obtained by the measuring device, which can be much larger than the eigenvalue. That is called the weak value amplification effect, which is also the principle for the measurement.
In this work, an initial state given by | sin | cos |i H Vψ α α= + was chosen as a pre-
selection state, where α was the angle between the first polarizer’s polarization axis and horizontal direction. According to Fresnel’s Formula, a refractive index-dependent phase difference Δ can be expressed as
2 21 sin 1
2 tansin tan
n
n
θθ θ
− −Δ = (1)
where θ was the incident angle and 1 2n n n= . n1 and n2 were refractive indices of the prism
and the analytes respectively. Next, the system to be detected interacted with the measuring
device weakly. The weak interaction in this experiment coupled between the polarization
operator and the longitudinal momentum then caused the spectrum to shift.
A second polarizer was used to prepare the post-selection state with its polarization axis
placed at angle of π⁄2+β with the first polarizer’s axis, where β was very small. The post-
selection state was given by ( )| cos( ) | sin( ) |if H e Vδψ α β α β Δ+= + + + , where Δ denoted
the refractive-index sensitive phase difference introduced by TIR [1], and δ was the phase
caused by the SBC. The weak value of the observable | | | |A H H V V= − can be
expressed as
( )
( )
ˆ| | sin cos( ) cos sin( )
sin cos( ) cos sin( )
if i
w if i
A eA
e
δ
δ
ψ ψ α α β α α βα α β α α βψ ψ
+Δ
+Δ
+ + +≡ =+ − + (2)
The imaginary part of the weak value was
( )
( ) 2
1 2 sin( )Im
1 1 2 cos( )
i
w i
eA
e
δ
δγ γ δγ γ γ δ
+Δ
+Δ
+ + Δ= ≈− + − + Δ
(3)
where = cot tan( )γ α α β+ . The shift of the momentum was given by 22 ( ) Im wP k P Aδ = Δ
[2], where ΔP was the uncertainty of the photons corresponding to the spectrum width of the
SLD, and k was the interaction strength [3], which was relatively small. We can rewrite δP
using the relation 2P π λ= , so the spectrum shift , as shown in Fig. 3a was
2 2 2
2 20 0
2 ( ) 4 ( ) sin( )= Im
(1 2 cos( ))W
k k aA
a
π δλ π δλ γ χλλ λ γ γ χ
+ Δ− = −+ − + Δ
(4)
2. The total internal reflection (TIR) experimental setup
In order to compare the resolution of the TIR and the weak measurement (WM), the TIR and the WM systems were built by the same experiment components for reducing the impact of equipment as much as possible as shown in Fig. S1. The two linear polarizers in Fig. S1a were nearly vertical to reduce light intensity.
Fig. S1. Thweak measu(SLD); B: adetection mTIR systemachromatic polarizer (Psystem.
3. The selfelt
A weak meagraphite felt wCV test and tPt017, Tianjinmm, purity: 9(D3 in Fig. S2test. The schreference electhe Fig. S2b.
The expandshown in Fig.was pressed aS2c) against water bath poFinally, the pby fixators (Dplatinum elecworking elect
he comparison of urement (WM) sysachromatic lens (L
module (EDM); F:m. (c) Schematic
lens (L1); C1: linP2); G1: achroma
f-made elec
asurement systwas rebuilt for the LSV test wn Aida Hengsh99.95%) was in2c, 10 × 10 × ematic view o
ctrode) system
ded view of th. S2c. A prism a graphite felt (an electrochemool (D9 in Figprism, the electD1 and D10 itrode (WE, D6trode, a graphit
the electrochemicstem. (a) Schemati
L1); C: linear polar achromatic lens of the electroch
near polarizer (P1atic lens (L2); H1
trochemical
tem with the CV testing as
was the workinheng Technolonserted from th5.4 mm3, com
of the three-elwithout the wa
he self-made e(D2 in Fig. S2
(D3 in Fig. S2cmical reaction g. S2c) to redutrochemical ren Fig. S2c) w
6 in Fig. S2c) wte rod (CE, D7
al total internal reic of the electrocherizer (P1); D: line(L2); G: spectrogr
hemical WM syst); D1: EDM; E1: : spectrograph. (d
detection m
electrochemicshown in Fig.
ng electrode. Aogy Developmhe middle of o
mpression ratio:ectrode (workater bath pool
electrochemica2c) was fixed wc) that was helcell (D5 in Fi
uce the interferaction cell and
with screws. Inwas inserted int7 in Fig. S2c) a
eflection (TIR) sysemical TIR system
ear polarizer (P2);raph. (b) A phototem. A1: SuperluSoleil-Babinet co
d) A photograph
module cont
cal detection mS2a. The only
A platinum elecment Co., LTDone side and th: 8%) as the w
king electrode,in the EDM in
al detection mwith a fixator (d by a graphiteig. S2c), whichrence caused bd the water ban the electrochto the graphite
and a saturated
stem and the electm. A: SuperlumineE: self-made elect
ograph of the electuminescent diode ompensator (SBC)of the electroche
taining the g
module contaiy difference betctrode (D6 in
D, China, ϕ 1 mhrough the gra
working electro, counter electn Fig. S2a was
module in Fig. (D1 in Fig. S2ce felt holder (Dh was surrounby outside temath pool were chemical reactioe felt (D3 in Fig
calomel electr
trochemical
escent diode trochemical trochemical (SLD); B1: ); F1: linear emical WM
graphite
ining the tween the Fig. S2c, mm × 37 aphite felt ode in CV trode and shown in
S2a was c), then it
D4 in Fig. nded by a mperature. combined on cell, a g. S2c) as rode (RE,
D8 in Fig. S2three-electrodthree-electrod
Fig. S2. (a) Sch(SLD); B: achroBabinet compensthree-electrode sfixator; D2: priselectrode (WE, wreference electrodthe RE and the w
4. Combin
As shown in tthrough the traditional cocombination shas some advpreferable. (2(3) the conduuniform currebetween the pexperiment.
2c) used as coude system was de system, the C
hematic of the elecmatic lens; C: linsator (SBC); F: liystem without the
sm; D3: graphite working electrode)de); D9: water bat
water bath pool in t
nation of the
the Fig. S3, thegraphite felt a
ombination strstructure betwevantages [4]: ) the platinum
uctivity of platient distributionplatinum electr
unter electrode shown in Fig
CE, the RE and
ctrochemical WMnear polarizer; D: inear polarizer; Ge water bath poolfelt; D4: graphite
); D7: graphite rodth pool; D10: fixathe EDM.
platinum ele
e platinum elecas the workinructure betweeeen the platinu(1) the corro
m wire has few inum is high, ns over the gode and the gr
and reference g. S2d. In orded the water bat
system containinself-made electro
G: achromatic lensl in the EDM. (c)e felt holder; D5d (CE, counter eletor. (d) The picture
ectrode and t
ctrode was inseng electrode inen the titaniuum electrode asion resistanceactive areas, w
so the combinraphite felt. T
raphite felt was
electrode resper to demonstrth pool were re
ng the graphite feltochemical detectios; H: spectrograph) Expanded schem: electrochemical ectrode); D8: satue of the three-elec
the graphite
erted from the in the CV tesum plate and and the graphite of platinumwhich has littl
nation structureTherefore, thiss used as the w
pectively. The prate the structuemoved in the E
t. A: Superlumineon module (EDM)h. (b) Schematic vmatic view of the
reaction cell; D6urated calomel electrode system with
felt
middle of onest. Compared
the graphite te felt mention
m in acid electe influence one can provide rs combination working electro
picture of ure of the EDM.
escent diode ); E: Soleil-view of the EDM, D1:
6: platinum ctrode (RE,
hout the CE,
e side and with the felt, the
ned above trolyte is
n CV test. relatively structure
ode in our
F
References
1. R. M. A. AzOpt. Soc. A
2. N. W. M. RLett. 66,110
3. X. Y. Xu, YMeasureme
4. L. T. Wu, Jbattery elec
Fig. S3. Schematic
zzam, “Phase shiftAm. A 21, 1559 (20Ritchie, J. G. Story07 (1991). Y. Kedem, K. Sun,ent Using a White . S. Wang, Y. She
ctrodes,” Phys. Che
c of the combinatio
fts that accompany004).
y, and R. G. Hulet,
, L. Vaidman, C.-FLight Source,” Phn, L. Liu and J. Yem. Chem. Phys. 1
on of the platinum
total internal refle
“Realization of a
F. Li, and G.-C. Guhys. Rev. Lett. 111. Xi, “Electrochem19, 14708-14717 (
m electrode and the
ection at a dielectr
Measurement of a
uo, “Phase Estima, 033604 (2013).
mical evaluation m(2017).
e graphite felt.
ric-dielectric interf
a "Weak Value",”
ation with Weak
methods of vanadiu
face,” J.
Phys. Rev.
um flow
